The Stacks project

35.20 Germs of schemes

Definition 35.20.1. Germs of schemes.

  1. A pair $(X, x)$ consisting of a scheme $X$ and a point $x \in X$ is called the germ of $X$ at $x$.

  2. A morphism of germs $f : (X, x) \to (S, s)$ is an equivalence class of morphisms of schemes $f : U \to S$ with $f(x) = s$ where $U \subset X$ is an open neighbourhood of $x$. Two such $f$, $f'$ are said to be equivalent if and only if $f$ and $f'$ agree in some open neighbourhood of $x$.

  3. We define the composition of morphisms of germs by composing representatives (this is well defined).

Before we continue we need one more definition.

Definition 35.20.2. Let $f : (X, x) \to (S, s)$ be a morphism of germs. We say $f$ is ├ętale (resp. smooth) if there exists a representative $f : U \to S$ of $f$ which is an ├ętale morphism (resp. a smooth morphism) of schemes.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04QQ. Beware of the difference between the letter 'O' and the digit '0'.